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• W2157303862 abstract "The problem of determining the structure of Nk(G) is of interest in several contexts. The counting problem goes back to Philip Hall, who expressed Nk(G) as a Mobius type summation of Nk(H) over all maximal subgroups H ⊂ G (see [23]). Recently the counting problem has been studied for large simple groups where remarkable progress has been made (see [25, 27]). In this paper we analyze Nk for solvable groups and products of simple groups. The sampling problem, while often used in theory as a tool for approximate counting, recently began a life of its own. In this paper we will present an algorithm for exact sampling in case when G is nilpotent. When little about structure of G is known, one can only hope for approximate sampling. In [11] Celler et al. proposed a product replacement Markov chain on Nk(G) which is conjectured to be rapidly mixing to a uniform stationary distribution. The subject was further investigated in [6, 12, 17, 16], while the conjecture is fully established only when G ' Zp, p is a prime. We prove rapid mixing for all abelian groups G. Also, we disprove the folklore conjecture that the group elements in generating k-tuples are (nearly) uniformly distributed. Finally, we would like to remark that the generating k-tuples occur in connection with the so-called random random walks, which are the ordinary random walks on G with random support. The analysis of these “average case” random walks was inspired by Aldous and Diaconis in [1] and was continued in a number of papers (see e.g. [19, 33, 36, 39]). We explain how the sampling problem can be used to test convergence of random random walks." @default.
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• W2157303862 date "1999-07-01" @default.
• W2157303862 modified "2023-09-26" @default.
• W2157303862 title "On sampling generating sets of finite groups and product replacement algorithm" @default.
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• W2157303862 doi "https://doi.org/10.1145/309831.309871" @default.
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